Cobordism of polarized t-structures

Abstract

Polarized t-structures seem to be attractive topological objects. One readily sees how to define the notion of the cobordism between them. This cobordism group Qrr (in dimension 3) is our principal object of interest in this paper. When one looks for cobordism invariants from this perspective, the following idea comes to mind first: The family g, gives rise to a metric on M x R+, as explained in Cl]. The conditions l-3 imply that the sectional curvature of this metric is bounded by one, and the volume is finite. Filling M with an arbitrary compact manifold W, and extending the metric to the filling we obtain a complete, finite volume and bounded curvature metric on the interior of W which is equal to the Cheeger-Gromov metric in the neighborhood of infinity. One can show that the integrals over W of the Pontrjagin forms computed from the metric, reduced mod 2 are cobordism invariants of the polarized t-structure. In fact complete open manifolds of bounded curvature and finite volume give rise to an interesting cobordism theory, and the integrals of Pontrjagin forms supply us with cobordism invariants. It is unclear just how well do these integrals gauge the cobordism group. In the present paper we construct, in the restricted context of polarized t-structures, finer invariants, which in fact allow the computation of the 3-dimensional cobordism group .$“I modulo torsion. We then use an invariant related to the integral of the Pontrjagin form to deal with the torsion.